Mathematics 101 Transitional Mathematics

Study Guide :: Unit 4

Polynomials and Rational Functions

Algebra, as we know it today, was more or less first formalized by Mohammad al-Khwarizmi (circa 780–850), a Persian mathematician, whose published text, The Science of Reunion and Reduction, systematically worked through solutions of linear and quadratic equations.

Some eight or so centuries later, René Déscartes (circa 1637) introduced what I call the ${\displaystyle x}$-factor into algebra by putting a grid onto what was previously the sand-strewn abakon of the Ancient Greeks. In doing so, he revolutionized the study of geometry. Once drawn on a planar tablet with compass, ruler and protractor, lines, parabolas, ellipses and other interesting planar curves could now be described primarily by their equations in the variables ${\displaystyle x}$ and ${\displaystyle y}$ in a coordinatized plane rather than by their descriptive geometric properties. It was found that both planar curves and traces of paths in 3-dimensional space were all expressible in terms of what we now call algebraic polynomial equations in one or more variables, thereby opening the door to the investigation of polynomials of varying degrees.

This ultimately led to a generalized definition of polynomials and the study of their properties as functions, a term first coined by one of the inventors of Calculus, Gottfried Leibniz, in 1673. However, it was Leonard Euler (1707–1783), a Swiss mathematician, who introduced the notation ${\displaystyle f(x)}$ to represent the value of a function, called ${\displaystyle f}$, at the real number ${\displaystyle x}$. In these tutorials, we often use either ${\displaystyle p(x)}$ or ${\displaystyle q(x)}$ to represent polynomial functions and their values.

Rational functions evolved out of the study of polynomials. Because we can divide functions, we can form the quotient of two polynomials to obtain what is called a rational function. The term ‘rational,’ of course, refers to the fact that it is a fraction or a ratio. Recall that the entire set of integer fractions is referred as the set of rational numbers.

It should be noted that every polynomial ${\displaystyle p(x)}$ may be considered a rational function ${\displaystyle r(x)=\frac{p(x)}{q(x)}\equiv \frac{p(x)}{1}}$ whose denominator polynomial ${\displaystyle q(x)}$ is the constant monomial 1. Therefore, all techniques of solving rational equations and inequalities apply equally to the solutions of polynomial equations and inequalities.

Learning Objectives

After you have completed Unit 4, you should be able to:

1. define and give examples of polynomials and rational functions;
2. list the fundamental properties of polynomials and rational functions to any base, calculate their values and to be able to perform operations on them as algebraic expressions;
3. factor both polynomials and rational functions;
4. solve equations and inequalities involving both polynomials and rational functions;
5. apply polynomials and rational functions to real-world situations;
6. determine the domains, ranges, fundamental properties, values and graphs of polynomials and rational functions; and
7. state the Remainder Theorem and the Factor Theorem for polynomials and apply these theorems to them.

Readings and Practice

Chapter 4 (pp. 95–154) of the textbook

Optional Section

• Focus on Modeling: fitting polynomial curves to data (pp. 155–159)

Online graphing tool: Graph

Quadratic Functions and Models

Textbook Readings

pp. 95–101

Practice

(pp. 101–104): 1, 3, 5, 9, 13, 17, 23, 27, 33, 35, 45, 51, 63, 69

Answers

pp. 502–503

Terms to Understand

a quadratic polynomial as a quadratic function, maximum / minimum value of a quadratic function

Figures/Tables

Definition of Quadratic Function (p. 96)
Standard Form of a Quadratic Function (p. 96)
Minimum or Maximum Value of a Quadratic Function (pp. 97, 99)

Online Tutorials—the Overview

Pertinent to “Quadratic Functions and Models”

• Polynomials: an introduction
• Parabolas in the real world: as quadratic equations in two variables ${\displaystyle x}$ and ${\displaystyle y}$
• Parabolas and their properties
• Parabolas and their equations

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Polynomial Functions and Their Graphs

Textbook Readings

pp. 104–114

Optional Subsections

• Shape of the Graph Near a Zero (pp. 112–113)
• Local Maxima and Minima of Polynomials (definition and recognition only) (pp. 113–114)

Practice

(pp. 115–118): 5, 7, 9, 11, 15, 19, 21, 25, 27, 33, 41, 49, 59, 61, 63, 81

Answers

pp. 503–505

Terms to Understand

formal definitions of a polynomial and its terminology (terms, coefficients, constant term, linear term, quadratic term, cubic term, leading term, leading coefficient, degree); graph of a polynomial, end behaviour of a polynomial; zeroes of a polynomial; local maxima and local minima of a polynomial

Figures/Tables

Definition and Terminology of Polynomial Functions (p. 104)
End Behaviour of Polynomials (p. 107)
Real Zeroes of Polynomials (p. 108)
Guidelines for Graphing Polynomial Functions (p. 109)

Online Tutorials—the Overview

Pertinent to “Polynomial Functions and Their Graphs”

• Polynomials: an introduction
• Arithmetic Operations on Polynomials: addition, subtraction and multiplication
• Polynomials: properties and graphs as functions Functions
  Examples (pages 1–11)
• Factoring Polynomials: linear, quadratic and higher order polynomials
  Polynomial Factoring: an introduction
  Common Factoring
  Factoring Quadratic Polynomials I
  Factoring Quadratic Polynomials II
  Factoring Quadratic Polynomials III
  Factoring Cubic Polynomials I
  Factoring Cubic Polynomials II

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Dividing Polynomials

Textbook Readings

pp. 118–123

Note: To divide one polynomial into another of equal or greater degree, use either the long division method or the synthetic division method. Knowledge of both is not required.

Practice

(pp. 123–124): 1, 3, 7, 9, 11, 17, 29, 39, 43, 47, 53, 59, 65

Answers

pp. 505

Terms to Understand

long division of polynomials and its associated terms dividend, divisor, quotient, and remainder; synthetic division of polynomials; Division Algorithm; Remainder Theorem; Factor Theorem

Figures/Tables

Division Algorithm (p. 119)
Remainder Theorem (p. 121)
Factor Theorem (p. 122)

Online Tutorials—the Overview

Pertinent to “Dividing Polynomials”

• None

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the ‘Polynomials and Factoring’ topic page.

Real Zeros of Polynomials (and how to solve polynomial equations)

Textbook Readings

pp. 125–128

Optional Subsections

• Descartes’ Rule of Signs and Upper and Lower Bounds for Roots (pp. 128–130)
• Using Algebra and Graphing Devices to Solve Polynomial Equations (pp. 130–131)

Practice

(pp. 132–135): 1, 3, 5, 7, 13, 15, 21, 27, 29, 35, 39, 47, 49, 99

Keep in mind that the Division Algorithm, the Remainder Theorem, the Factor Theorem, and the Quadratic Formula may be used to find real zeros of polynomials.

Answers

pp. 505–506

Terms to Understand

zero of a polynomial; root of a polynomial; solution set of a polynomial equation

Figures/Tables

Rational Zeros Theorem (p. 125)

Online Tutorials—the Overview

Pertinent to “Real Zeros of Polynomials”

Remember that the real zeros of polynomials are found by setting the factors of the polynomial to 0 and solving the resulting, mostly linear and quadratic, equations. You may use any of the polynomial factoring techniques to find the factors of certain low degree polynomials. However, in more complex and higher degree polynomials, the Rational Zeros Theorem may be necessary to find at least one of the rational zeros. From there, use the Remainder Theorem, the Factor Theorem, and long division to find the others. Also be aware that any techniques of solving rational equations and rational inequalities apply to polynomials because each polynomial may be viewed as rational function with the constant monomial 1 as its denominator.

• Solving Polynomial Equations and Inequalities
  Rational Functions
  Rational Equations
  Rational Inequalities

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Rational Functions

Textbook Readings

pp. 136–147

Practice

(pp. 148–154): 1–6, 7, 9, 11, 15, 19, 21, 23, 25, 33, 41, 45, 49, 59, 65, 73 using Graph, 83

Answers

p. 506

Terms to Understand

formal definition of a rational function; vertical, horizontal and slant asymptotes

Figures/Tables

Definition of Vertical and Horizontal Asymptotes (p. 138)
Finding Asymptotes of Rational Functions (p. 141)
Sketching Graphs of Rational Functions (p. 142)

Online Tutorials—the Overview

Pertinent to “Rational Functions”

• Rational Functions: definition and properties
• Operations on Rational Functions (Multiply and Divide)
• Operations on Rational Functions (Add and Subtract)
• Solving Rational Equations
• Solving Rational Inequalities I
• Solving Rational Inequalities II

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Unit 4 Review

(pp. 151–153): odd numbers

Answers

pp. 506–508

Unit 4 Test

(p. 154): 1–11

Answers

pp. 508–510

Figures/Tables

Formulas to Remember (pp. iv and v at the beginning of the text)